Mean Deviation

The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean. For a sample size , the mean deviation is defined by

(1)

where  is the mean of the distribution. The mean deviation of a list of numbers is implemented in the Wolfram Language as MeanDeviation[data].

The mean deviation for a discrete distribution  defined for , 2, ...,  is given by

(2)

Mean deviation is an important descriptive statistic that is not frequently encountered in mathematical statistics. This is essentially because while mean deviation has a natural intuitive definition as the "mean deviation from the mean," the introduction of the absolute value makes analytical calculations using this statistic much more complicated than the standard deviation

(3)

As a result, least squares fitting and other standard statistical techniques rely on minimizing the sum of square residuals instead of the sum of absolute residuals.

For example, consider the discrete uniform distribution consisting of  possible outcomes with  for , 2, ..., . The mean is given by

(4)

The variance (and therefore its square root, namely the standard deviation) is also straightforward to obtain as

(5)

On the other hand, the mean deviation is given by

(6)

This can be obtained in closed form, but is much more unwieldy since it requires breaking up the summand into two pieces and treating the cases of  even and odd separately.

The following table summarizes the mean absolute deviations for some named continuous distributions, where  is an incomplete beta function,  is a beta function,  is a gamma function,  is the Euler-Mascheroni constant,  is a Meijer G-function,  is the exponential integral function,  is erf, and  is erfc.

distribution	M.D.
beta distribution
chi-squared distribution
exponential distribution
gamma distribution
Gumbel distribution
half-normal distribution
Laplace distribution
logistic distribution
log normal distribution
Maxwell distribution
normal distribution
Pareto distribution
Rayleigh distribution
Student's t-distribution
Student's t-distribution
triangular distribution
triangular distribution
uniform distribution

The following table summarizes the mean absolute deviations for some named discrete distributions, where .

distribution	M.D.
Bernoulli distribution
binomial distribution
discrete uniform distribution
geometric distribution
Poisson distribution
Zipf distribution

REFERENCES:

Havil, J. "Ways of Means." §13.1 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 119-121, 2003.

Kenney, J. F. and Keeping, E. S. "Mean Absolute Deviation." §6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 76-77 1962.